The braid group : redefining

The role of the braid group constitutes one of the invariant measurements. Through the classification of braids formed several parts of the braid group, but does not computationally distinguish them. Some characteristics have been expressed to give the features to a braid in braid group based on redefining relation between braid group and permutation group.


Introduction
One of technologies that has been growing ever since is the braid [1,2].The braid is widely used in everyday life [3].The braid is not only used by the public in general, but the academics, especially the biologists for genetic engineering [4].As a technology, the braid has its own mathematical model [5].The modelling is to express the privilege of a braid.
In group theory, the braid is one of the most interesting implementations [6], which provides a form of the invariant measurement about a structure [5].Thus, a braid has the general and special characteristics, which characterizes each braid so as to give the idea of meaning to the structure formed.Therefore, this paper redefines relations between groups based on the braids in the braid groups specifically and expresses the computational characteristics associated with it.

Woven and braid
Three woven patterns are σi 1 , σi 0 , and σi -1 each like Fig. 1.They build a system if there are two parallel lines L1 and L2 (can be hidden/not declared) in the same direction in the space R 3 , and in the line L1 there is the points pi, while in the line L2 there is points qi, where i, j = 1,…,n so that on each of the lines the distance of adjacent points is same, and then one point pi is connected by curve ci just to a point qj.L1 and L2 are called upper frame and bottom frame, respectively.σi, i = 1,…,n are generators [7].Computationally, Fig. 1.Triple of woven patterns in braid Notation σi 1 or simply written σi is a woven geometric shape where the curve ci overpass ci+1 exactly once on it, whereas cj for j ≠ i, i+1 directly connects the points pj to qj.Instead σi -1 is a woven where the curve ci underpass ci-1 exactly once on it, whereas cj, j ≠ i-1, i is direct connect the points pj to qj, see Fig.
2 [3].The arrangement of points in a pair of frame lines can be expressed as a permutation form as follows c is the curve that maps the points at L1 to L2. Number of generator σ for representing A is more than one woven (minimally is σ 0 ), and we denote it as nσ().For example, for i = 1,2,…,6 we have one woven arrangement A1 = σ3 -1 σ2σ4σ1 -1 σ3 -1 σ5σ2 and nσ(A1) = | σ3 -1 σ2σ4σ1 -1 σ3 -1 σ5σ2| = 7, see Fig.A braid is a woven together with a deformation operation where no woven being of same height that https://doi.org/10.1051/matecconf/201819701005AASEC 2018 during the deformation: (i) L1 and L2 remain parallel; (ii) There are no wedges between the curves; and (iii) The curve remains normal.The deformation can be expressed as a brushing, i.e. the projection of the curve ci moving from pi to qj has the distance from the line L1 will always increase.

Basic of braid group
A set A ≠ Ø together with binary operation • denoted by <A,•> is called a braid group if and only if: Binary operation between same ordered-braids can be expressed as a system [3], i.e. let L1,L2,pi,qj,ci and L1',L2',pi',qi',ci' respectively as frames, points, and curves of two braids A in A and A' in A, the braid A•A' (or AA') can be obtained through the union of L2 and L1' such that qj coincides with pi' and AA' in A is a new braid by removing L2 = L1'; Thus, for every same ordered-braid, the associative law applicable, i.e. (A1A2)A3 = A1(A2A3), A1, A2, A3 in A; Geometrically an identity braid, denoted by A 0 , can be expressed as a woven arrangement by which each curve ci connects pi directly to qj and i = j, so that when the identity braid composed with any other same orderedbraid, the resulted braid will be equal to the origin braid.While, geometrically the inverse of A can be guaranteed to exist through the results of reflection of A to the line L2 of A so that L2 becomes L1' of the braid A -1 , it constructed by the shadow of L1 against L2 whereby the curve ci' will be the inverse of the curve ci and consequently the woven on A where the curve ci overpass become the curve ci underpass A -1 , and their distance to L2 and L1' be equal.That is AA -1 = A 0 = A -1 A and we conclude that A 0 = 1 where A 0 in A and A -1 in A [7,8].
The class of same ordered braids together with the operation is a group.This group is not determined by the length of the curve ci of each braid, but only by the woven structure of σ in braids.We denoted An as a group of braids order n.However, the set of permutations that represent all the braids is also a group called the nordered permutation group and denoted by Pn [9].Depending on any arrangement of σj can be added/ inserted to any braid provided that the σj arrangement applies to j < i while the other curve of ci connects the same point.Thus, the adding process of the σj arrangement at the beginning or end of the braid is like performing a binary operation, whereas insertion is done by constructing a new line to cut and then adding new ones and combining other pieces based on appropriate binary operations.
Addition and insertion of structures such as Eq. ( 6) and Eq. ( 7) and their derivatives generally change the arrangement pattern in permutations that are an element of the Pn group.Therefore, there is a relation between two groups, An and Pn, as the classifications of elements in braid group.

An Approach
As has been expressed in its definition and characteristics, the woven arrangement that forms the braid gives a special meaning to each braid.Therefore, the braid group can be represented by two representations of structure and frame, the presence of the relation between two forms is a linkage that gives a special meaning.Thus, an approach is used to express the meaning of the structure and frames of the braid and relation between them.It is general expressed by group theory that there is a classification between braids that have the same meaning or in vary [11].
A mapping from the braid group An to the Pn permutation group is to define the classification of braids into appropriate classes [9], i.e. h : An → Pn (8) then the specific meaning is based on its structure through the composition of the generator that builds a braid.Disclosure of σ with index composition will be different meaning in the presence of σ -1 as opposed to.The ratio includes a comparison that assesses the braid structure.Likewise with the permutation groups, the same point ratio of positions on both frames connected by the same curve ci would differ in meanings of its MATEC Web of Conferences 197, 01005 (2018) https://doi.org/10.1051/matecconf/201819701005AASEC 2018 presence compared to the presence of curve ci connecting the different points between the two frame lines [3].Thus, it is a comparison about the structures or frames of braids and it gives the meaning of the existence of elements in braid group.

The Structure of Braid
Let σi is one generator of braids whereby its permutation is but for σi 2 the permutation is and we conclude that for σi p whereby p is even or p = 0,2,4,… we have their permutations are like Eq. ( 10), while p is odd or p = 1,3,5,… we have their permutations are like Eq. ( 9).The composition of generator is not only based on one index but a sequential index as σiσi+1σi+2… [8,3].Therefore, we have a lemma as follows.
Lemma 2. For each braid in a braid group, if there is one frame line on the braid then the line cuts the braid be two braid in a braid group.
Proof.Let A in A, the line cut the braid right on the σi 0 for i = 1,2,…,n so that there two braids: A1,A2 in A or A 0 ,A in A, we call them as the resultant braids, and be in force , whereby ((i)) is presentation of the normal braids.
Corollary 1.The insertion of un-normal braids to any other braid alters its permutation.
Proof.This is as consequence of Theorem 1 to Proposition 1.

Defining characteristics
Through Lemma 1, Lemma 2, Proposition 1 and Theorem 1 has been shown that there is a function h from An into Pn, that is which classifies each element of An into the same class based on elements of Pn.
Computationally, it can be declared some scales of calculation in braid group An, i.e.
[S1] A number of generators allowed in the braid group An, we denoted it as nn-1(), for example the braid group A6 has n6 = 5, or generally the braid group An based on definition of braid group having nn() = n-1.
To insert any braid into another braid is by first considering the curves ci connecting two frame lines.If both braids are in the same order, then the insertion can be done.If the number of curves ci of the inserted braid is smaller than the number of curves cj of braid accept the insertion, then the number of curves ci is adjusted to the number of curves cj by adding curves in the appropriate part.Also it applies vice versa.Therefore, based on Lemma 2, insertion can be done by separating one braid into two braids, and then carrying out binary operations for the three braids by order first braid, the inserted braid, and the last braid.